A Priori Estimates and Existence of Positive Solutions for a Quasilinear Elliptic Equation

On the basis of some new Liouville theorems, under suitable conditions, a priori estimates are obtained of positive solutions of the problem − Δ p u = λ u α − a (x) u q i n Ω, u | ∂ Ω = 0, where Ω ⊂ RN (N⩾ 2) is a bounded smooth domain, p>1 and λ is a parameter, α, q are given constants such that p−1 p and p*=∞ N ⩽ p, and a (x) is a continuous nonnegative function. Making use of the Leray–Schauder degree of a compact mapping and a priori estimates, the paper finds that the problem above possesses at least one positive solution. It also discusses the corresponding perturbed problem, where a (x) is replaced by a (x) +ε, ε>0. The results are strikingly different from those obtained for the case α=p−1.